N. 20, January 2002: Mr. Bayes and the true nature of scientific hypotheses

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Massimo gives us an overview of Bayesian statistics, as compared to the more well-known "frequentist" sort.

How does science work, really? You can read all about it in plenty of texts in philosophy of science, but if you have ever experienced the making of science on an everyday basis, chances are you will feel dissatisfied with the airtight account given by philosophers. Too neat, not enough mess.

To be sure, I am not denying the existence of the scientific method(s), as radical philosopher Paul Feyerabend is infamously known for having done. But I know from personal experience that scientists don’t spend their time trying to falsify hypotheses, as Karl Popper wished they did. By the same token, while occasionally particular scientific fields do undergo periods of upheaval, Thomas Kuhn’s distinction between “normal science” and scientific “revolutions” is too simple. Was the neo-Darwinian synthesis of the 1930s and 40s in evolutionary biology a revolution or just a significant adjustment? Was Eldredge and Gould’s theory of “punctuated equilibria” to explain certain features of the fossil record a blip on the screen or, at least, a minor revolution?

But, perhaps, the least convincing feature of the scientific method is not something theorized by philosophers, but something actually practiced by almost every scientist, especially those involved in heavily statistical disciplines such as organismal biology and the social sciences. Whenever we run an experiment, we analyze the data in a way to verify if the so-called “null hypothesis” has been successfully rejected. If so, we open a bottle of champagne and proceed to write up the results to place a new small brick in the edifice of knowledge.

Let me explain. A null hypothesis is what would happen if nothing happened. Suppose you are testing the effect of a new drug on the remission of breast cancer. Your null hypothesis is that the drug has no effect: within a properly controlled experimental population, the subjects receiving the drug do not show a statistically significant difference in their remission rate when compared to those who did not receive the drug. If you can reject the null, this is great news: the drug is working, and you have made a potentially important contribution toward bettering humanity’s welfare. Or have you?

The problem is that the whole idea of a null hypothesis, introduced in statistics by none other than Sir Ronald Fisher (the father of much modern statistical analyses), constrains our questions to ‘yes’ and ‘no’ answers. Nature is much too subtle for that. We probably had a pretty good idea, before we even started the experiment, that the null hypothesis was going to be rejected. After all, surely we don’t embark in costly (both in terms of material resources and of human potential) experiments just on the whim of the moment. We don’t randomly test all possible chemical substances for their role as potential anti-carcinogens. What we really want to know is if the new drug performed better than other, already known, ones — and by how much. That is, every time we run an experiment we have two factors that Fisherian (also known as “frequentist,” see below) statistics does not take into account: first, we have a priori expectations about the outcome of the experiments, i.e., we don’t enter the trial as a blank slate (contrary to what is assumed by most statistical tests); second, we normally compare more than two hypotheses (often several), and the least interesting of them is the null one.

An increasing number of statisticians and scientists are beginning to realize this, and are ironically turning to a solution that was devised, and widely used, well before Fisher. That solution was contained in an obscure paper that one Reverend Thomas Bayes published back in 1763, and is revolutionizing how scientists do their work, as well as how philosophers think about science.

Bayesian statistics simply acknowledge that what we are really after is an estimate of the probability of a certain hypothesis to be true, given what we know before running an experiment, as well as what we learn from the experiment itself. Indeed, a simple formula known as Bayes theorem says that the probability that a hypothesis (among many) is correct, given the available data, depends on the probability that the data would be observed if that hypothesis were true, multiplied by the a priori probability (i.e., based on previous experience) that the hypothesis is true.

In Fisherian terms, the probability of an event is the frequency with which that event would occur given certain circumstances (hence the term “frequentist” to identify this classical approach). For example, the probability of rolling a three with one (unloaded) die is 1/6, because there are six possible, equiprobable outcomes, and on average (i.e., on long enough runs) you will get a three one time every six.

In Bayesian terms, however, a probability is really an estimate of the degree of belief (as in confidence, not blind faith) that a researcher can put into a particular hypothesis, given all she knows about the problem at hand. Your degree of belief that threes come out once every six rolls of the die comes from both a priori considerations about fair dice, and the empirical fact that you have observed this sort of event in the past. However, should you witness a repeated specified outcome over and over, your degree of belief in the hypothesis of a fair die would keep going down until you strongly suspect foul play. It makes intuitive sense that the degree of confidence in a hypothesis changes with the available evidence, and one can think of different scientific hypotheses as competing for the highest degree of Bayesian probability. New experiments will lower our confidence in some hypotheses, and increase the one in others. Importantly, we might never be able to settle on one final hypothesis, because the data may be roughly equally compatible with several alternatives (a frustrating situation very familiar to any scientist and known in philosophy as the underdetermination of hypotheses by the data).

You can see why a Bayesian description of the scientific enterprise — while not devoid of problems and critics — is revealing itself to be a tantalizing tool for both scientists, in their everyday practice, and for philosophers, as a more realistic way of thinking about science as a process.

Perhaps more importantly, Bayesian analyses are allowing researchers to save money and human lives during clinical trials because they permit the researcher to constantly re-evaluate the likelihood of different hypotheses during the experiment. If we don’t have to wait for a long and costly clinical trial to be over before realizing that, say, two of the six drugs being tested are, in fact, significantly better than the others, Reverend Bayes might turn out to be a much more important figure in science than anybody has imagined over the last two centuries.

Many thanks to Melissa Brenneman and Bob Faulkner for patiently editing and commenting on Rationally Speaking columns.

Quote of the month:

A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it.

— Maxwell Planck

Further Reading:

Bayes or Bust?, by John Earman. Earman (a professor of History and Philosophy of Science at the University of Pittsburgh) argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianism fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns.

Web Links:

A collection of Bayesian sites to find software, theory, and discussions.

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